Theorem rmoeqi 36166

Description: Equality inference for restricted at-most-one quantifier. (Contributed by GG, 1-Sep-2025.)

Hypothesis

Ref	Expression
rmoeqi.1	⊢ 𝐴 = 𝐵

Assertion

Ref	Expression
rmoeqi	⊢ (∃*𝑥 ∈ 𝐴 𝜓 ↔ ∃*𝑥 ∈ 𝐵 𝜓)

Proof of Theorem rmoeqi

Step	Hyp	Ref	Expression
1		rmoeqi.1	. . . . 5 ⊢ 𝐴 = 𝐵
2	1	eleq2i 2832	. . . 4 ⊢ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)
3	2	anbi1i 624	. . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ (𝑥 ∈ 𝐵 ∧ 𝜓))
4	3	mobii 2547	. 2 ⊢ (∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜓) ↔ ∃*𝑥(𝑥 ∈ 𝐵 ∧ 𝜓))
5		df-rmo 3379	. 2 ⊢ (∃*𝑥 ∈ 𝐴 𝜓 ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜓))
6		df-rmo 3379	. 2 ⊢ (∃*𝑥 ∈ 𝐵 𝜓 ↔ ∃*𝑥(𝑥 ∈ 𝐵 ∧ 𝜓))
7	4, 5, 6	3bitr4i 303	1 ⊢ (∃*𝑥 ∈ 𝐴 𝜓 ↔ ∃*𝑥 ∈ 𝐵 𝜓)

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2108  ∃*wmo 2537  ∃*wrmo 3378

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-mo 2539  df-cleq 2728  df-clel 2815  df-rmo 3379

This theorem is referenced by:  disjeq1i  36171